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July 2010 Invariance principles for local times at the maximum of random walks and Lévy processes
L. Chaumont, R. A. Doney
Ann. Probab. 38(4): 1368-1389 (July 2010). DOI: 10.1214/09-AOP512


We prove that when a sequence of Lévy processes X(n) or a normed sequence of random walks S(n) converges a.s. on the Skorokhod space toward a Lévy process X, the sequence L(n) of local times at the supremum of X(n) converges uniformly on compact sets in probability toward the local time at the supremum of X. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and descending) converges jointly in law toward the ladder processes of X. As an application, we show that in general, the sequence S(n) conditioned to stay positive converges weakly, jointly with its local time at the future minimum, toward the corresponding functional for the limiting process X. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law.


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L. Chaumont. R. A. Doney. "Invariance principles for local times at the maximum of random walks and Lévy processes." Ann. Probab. 38 (4) 1368 - 1389, July 2010.


Published: July 2010
First available in Project Euclid: 8 July 2010

zbMATH: 1210.60033
MathSciNet: MR2663630
Digital Object Identifier: 10.1214/09-AOP512

Primary: 60F17
Secondary: 60G17 , 60J15 , 60J55

Keywords: invariance principle , ladder processes , local time at the supremum , Meander , processes conditioned to stay positive

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 4 • July 2010
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